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Ch. 5 - Trigonometric Identities
Lial - Trigonometry 12th Edition
Lial12th EditionTrigonometryISBN: 9780136552161Not the one you use?Change textbook
All textbooksLial 12th EditionCh. 5 - Trigonometric IdentitiesProblem 5.1.62
Chapter 6, Problem 5.1.62

Write each expression in terms of sine and cosine, and then simplify the expression so that no quotients appear and all functions are of θ only. See Example 3.
(sec θ - 1) (sec θ + 1)

Verified step by step guidance
1
Recall the definition of secant in terms of cosine: \(\sec \theta = \frac{1}{\cos \theta}\).
Rewrite the expression \((\sec \theta - 1)(\sec \theta + 1)\) by substituting \(\sec \theta\) with \(\frac{1}{\cos \theta}\), so it becomes \(\left(\frac{1}{\cos \theta} - 1\right) \left(\frac{1}{\cos \theta} + 1\right)\).
Recognize that the expression is a product of conjugates, which follows the difference of squares formula: \((a - b)(a + b) = a^2 - b^2\). Here, \(a = \frac{1}{\cos \theta}\) and \(b = 1\).
Apply the difference of squares formula to get \(\left(\frac{1}{\cos \theta}\right)^2 - 1^2 = \frac{1}{\cos^2 \theta} - 1\).
Rewrite the expression \(\frac{1}{\cos^2 \theta} - 1\) as a single fraction with denominator \(\cos^2 \theta\), resulting in \(\frac{1 - \cos^2 \theta}{\cos^2 \theta}\), and then use the Pythagorean identity \(\sin^2 \theta = 1 - \cos^2 \theta\) to express the numerator in terms of sine.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Definition of Secant in Terms of Cosine

Secant (sec θ) is the reciprocal of cosine, defined as sec θ = 1/cos θ. Expressing secant in terms of cosine allows rewriting expressions involving secant into sine and cosine, which are the fundamental trigonometric functions.
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Graphs of Secant and Cosecant Functions

Algebraic Simplification of Trigonometric Expressions

Simplifying trigonometric expressions involves combining like terms, factoring, and eliminating quotients by multiplying numerator and denominator appropriately. This process helps rewrite expressions without fractions and in terms of sine and cosine only.
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Pythagorean Identity

The Pythagorean identity states that sin²θ + cos²θ = 1. This identity is essential for simplifying expressions by replacing sin²θ or cos²θ terms, enabling the expression to be written in a simpler form involving only sine and cosine.
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Related Practice
Textbook Question

For each expression in Column I, choose the expression from Column II that completes an identity. One or both expressions may need to be rewritten.

sec x/csc x


II

A. sin ^2 x/cos ^2 x

B.1/(sec ^2 x)

C. sin (-x)

D. csc ^2 x-cot ^2 x + sin ^2 x

E. tan x

Textbook Question

Verify that each equation is an identity.

(tan² α + 1)/ sec α = sec α

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Textbook Question

Find the exact value of each expression. (Do not use a calculator.)

cos(-15°)

Textbook Question

Write each expression in terms of sine and cosine, and then simplify the expression so that no quotients appear and all functions are of θ only. See Example 3.

cot² θ(1 + tan² θ)

Textbook Question

Verify that each equation is an identity.

(sec α - tan α)² = (1 - sin α)/(1 + sin α)

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Textbook Question

Verify that each equation is an identity.

sin² β (1 + cot² β) = 1

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